Page:Scientific Papers of Josiah Willard Gibbs.djvu/243

Rh $$\frac{dx}{dx'}, ... \frac{dz}{dz'}$$ under the most general supposition with respect to the position of the co-ordinate axes.

For any principal axis of strain we have  the differential coefficients in the first of these equations being determined from (420) as before. Therefore, From (420) we obtain directly  From the two last equations, in virtue of the necessary relation $$\alpha^{'2} + \beta^{'2} + \gamma^{'2} = 1$$, we obtain  or, if we substitute the values of the differential coefficients taken from (420),

If we eliminate $$\alpha ', \beta ', \gamma '$$ from these equations, we may write the result in the form, We may write